cp's OEIS Frontend

A103670 Smallest m such that A102730(m) = n.

%I A103670 #26 Apr 16 2025 19:38:24
%S A103670 0,1,2,3,4,5,8,23,117,155,1410,3702,29406,104312,680407
%N A103670 Smallest m such that A102730(m) = n.
%C A103670 _Reinhard Zumkeller_ conjectures (at A102730) that this sequence is finite. I conjecture the contrary, that a(n) exists for every n.  Further, I expect a(n) << n^n. - _Charles R Greathouse IV_, Aug 21 2011
%C A103670 a(14) > 41000, if it exists. - _Amiram Eldar_, Apr 03 2025
%C A103670 The number of times that 1! through 10! appear in a(11)! is: 17417, 8777, 4361, 1118, 265, 31, 5, 1, 1, 1. It suggests factorials are normal in base 2 except for their trailing zeros. On this assumption, a(16) is expected around 3-4 million, and a(17) around 20-30 million. - _Martin Fuller_, Apr 14 2025
%e A103670 a(6) = 5: A102730(5) = #{0,1,2,3,4,5} = 6.
%e A103670 a(7) = 8: A102730(8) = #{0,1,2,3,4,7,8} = 7.
%e A103670 a(8) = 23: A102730(23) = #{0,1,2,3,4,5,6,23} = 8.
%e A103670 a(9) = 117: A102730(117) = #{0,1,2,3,4,5,6,7,117} = 9.
%e A103670 a(10) = 155: A102730(155) = #{0,1,2,3,4,5,6,7,8,155} = 10.
%t A103670 A102730[n_] := Sum[Boole[StringContainsQ[IntegerString[n!, 2], IntegerString[k!, 2]]], {k, 0, n}];
%t A103670 seq[len_] := Module[{s = Table[0, {len}], c = 0, m = 0, i}, While[c < len, i = A102730[m]; If[i <= len && s[[i]] == 0, c++; s[[i]] = m]; m++]; s]; seq[10] (* _Amiram Eldar_, Apr 03 2025 *)
%Y A103670 Cf. A102730.
%K A103670 nonn,base,more,hard
%O A103670 1,3
%A A103670 _Reinhard Zumkeller_, Feb 12 2005
%E A103670 a(11)-a(12) from _D. S. McNeil_, Aug 21 2011
%E A103670 a(13) from _Amiram Eldar_, Apr 03 2025
%E A103670 a(14)-a(15) from _Martin Fuller_, Apr 14 2025